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<title> BALSAC Version 2.15 Manual, Section 5.3</title>
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<h1>5.3.  INTERNAL PARAMETERS FOR ATOMS, LATTICES</h1>

<a href="./balm.48.html">next</a>, <a href="./balm.46.html">previous</a> Section  /  <a href="./balm.0.html">Table of Contents</a>  /  <a href="./balm.127.html">Index</a>

 
For clusters and general lattices (codes 10,-10) values of atomic radii may
be determined as renormalized default radii according to respective nuclear
charges such that maximum packing without overlapping spheres is achieved,
see Secs. <a href="./balm.56.html">6.2.1</a>, <a href="./balm.83.html">6.3.2</a>. The internal table of default atomic radii (given in
Angstroms) is shown in the following.
 
  charge   radius    charge   radius    charge   radius    charge   radius
 -----------------  -----------------  -----------------  -----------------
   1 (H )  0.4350    25 (Mn)  1.3500    48 (Cd)  1.4894    71 (Lu)  1.7515
   2 (He)  1.4000    26 (Fe)  1.2411    49 (In)  1.6662    72 (Hf)  1.5973
   3 (Li)  1.5199    27 (Co)  1.2535    50 (Sn)  1.5375    73 (Ta)  1.4280
   4 (Be)  1.1430    28 (Ni)  1.2460    51 (Sb)  1.4000    74 (W )  1.3705
   5 (B )  0.9750    29 (Cu)  1.2780    52 (Te)  1.3600    75 (Re)  1.3800
   6 (C )  0.6550    30 (Zn)  1.3325    53 (I )  1.3300    76 (Os)  1.3676
   7 (N )  0.7500    31 (Ga)  1.3501    54 (Xe)  2.2000    77 (Ir)  1.3573
   8 (O )  0.7300    32 (Ge)  1.2248    55 (Cs)  2.6325    78 (Pt)  1.3873
   9 (F )  0.7200    33 (As)  1.2000    56 (Ba)  2.1705    79 (Au)  1.4419
  10 (Ne)  1.6000    34 (Se)  1.1600    57 (La)  1.8725    80 (Hg)  1.5025
  11 (Na)  1.8579    35 (Br)  1.1400    58 (Ce)  1.8243    81 (Tl)  1.7283
  12 (Mg)  1.6047    36 (Kr)  2.0000    59 (Pr)  1.8362    82 (Pb)  1.7501
  13 (Al)  1.4318    37 (Rb)  2.4700    60 (Nd)  1.8295    83 (Bi)  1.4600
  14 (Si)  1.1758    38 (Sr)  2.1513    61 (Pm)  1.8090    84 (Po)  1.4600
  15 (P )  1.0600    39 (Y )  1.8237    62 (Sm)  1.8040    85 (At)  1.4500
  16 (S )  1.0200    40 (Zr)  1.6156    63 (Eu)  1.9840    86 (Rn)  1.4300
  17 (Cl)  0.9900    41 (Nb)  1.4318    64 (Gd)  1.8180    87 (Fr)  2.5000
  18 (Ar)  1.9000    42 (Mo)  1.3626    65 (Tb)  1.8005    88 (Ra)  2.1400
  19 (K )  2.2620    43 (Tc)  1.3675    66 (Dy)  1.7951    89 (Ac)  1.8775
  20 (Ca)  1.9758    44 (Ru)  1.3529    67 (Ho)  1.7886    90 (Th)  1.7975
  21 (Sc)  1.6545    45 (Rh)  1.3450    68 (Er)  1.7794    91 (Pa)  1.6086
  22 (Ti)  1.4755    46 (Pd)  1.3755    69 (Tm)  1.7687    92 (U )  1.5683
  23 (V )  1.3090    47 (Ag)  1.4447    70 (Yb)  1.9396    93-100   1.0000
  24 (Cr)  1.2490
 
 
In addition to a free definition of lattices (lattice codes 10, -10),
structure parameters for 9 different standard lattices as well as for all 14
Bravais lattices are defined internally and can be selected by respective
lattice code numbers inside a LATTICE session. The following tables list all
internal parameters.
 
 
A) Lattice types
 
    -----------------------------------------------------------
      code (NTYP)   internal title    lattice type
    -----------------------------------------------------------
        1           sc                simple cubic
        2, -2       fcc               face centered cubic
        3, -3       bcc               body centered cubic
        4, -4       hcp (hex+2)       hexagonal closed packed
        5, -5       diamond(fcc+2)    diamond
        6, -6       NaCl (fcc+2)      sodium chloride
        7           CsCl (sc+2)       cesium chloride
        8, -8       Znblnde(fcc+2)    cubic Zincblende
        9, -9       Graphite          Graphite
       10, -10                        free lattice
       10           Bravais #  1      triclinic-P
       10           Bravais #  2      monoclinic-P
       10           Bravais #  3      monoclinic-C
       10           Bravais #  4      orthorhombic-P
       10           Bravais #  5      orthorhombic-C
       10           Bravais #  6      orthorhombic-I
       10           Bravais #  7      orthorhombic-F
       10           Bravais #  8      tetragonal-P
       10           Bravais #  9      tetragonal-I
       10           Bravais # 10      hexagonal-P
       10           Bravais # 11      trigonal-R
       10           Bravais # 12      cubic-P
       10           Bravais # 13      cubic-I
       10           Bravais # 14      cubic-F
    -----------------------------------------------------------
 
 
B) Lattice parameters
 
B1) Standard lattices
 
The following tables give, for each of the 9 standard lattices, cartesian
coordinates of all lattice vectors of the real and reciprocal lattice and
coordinates of all lattice basis vectors of non-primitive lattices. Also
shown are default nuclear charges and atom/ion radii. The radii are
calculated for touching spheres of maximum space filling and are used to
determine the packing ratio (defined as the relative amount of space filled
by the spheres). The point symmetry elements of the lattices reflect the
point symmetry groups
 
   the cubic group                 for sc, fcc,bcc, NaCl, CsCl lattices,
   the tetrahedral group           for diamond, cubic ZnS lattices,
   the reduced hexagonal group     for hcp, graphite lattices.
 
The symmetry labels and vectors (vx,vy,vz) are defined as
 
  "1"    for identity operation with (vx,vy,vz) denoting the lattice origin,
  "I"    for inversion operation with (vx,vy,vz) denoting the lattice origin,
  "Cn"   for n-fold rotational axes with (vx,vy,vz) pointing along the axis
         (only n = 1, 2, 3, 4, 6 are meaningful),
  "Mn"    for n-fold mirror planes with (vx,vy,vz) pointing along the mirror
          plane normal. Here n = 2 denotes standard mirror planes while
          n = 3, 4, 6 refer to n-fold rotations combined with inversions.
 
In the following all coordinates in real space are given as multiples of the
lattice constant A, those of the reciprocal space as multiples of 1/A. The
numerical values are taken from BALSAC output (all parameters are defined
internally with full REAL*8 accuracy).
 
a)  simple cubic lattice (lattice code = 1).
 
  lattice vectors                    reciprocal lattice vectors
         x         y         z               x         y         z
  R1   1.00000   0.00000   0.00000    G1   6.28319   0.00000   0.00000
  R2   0.00000   1.00000   0.00000    G2   0.00000   6.28319   0.00000
  R3   0.00000   0.00000   1.00000    G3   0.00000   0.00000   6.28319
 
  1 lattice basis vector
          x         y         z      atom/ion radius   nuclear charge
  r1   0.00000   0.00000   0.00000       0.5000            1   (H)
 
  packing ratio =  0.52360
 
  48 symmetry operations :  1x1, 9xC2, 8xC3, 6xC4, 1xI, 9xM2, 8xM3, 6xM4
 
    type(s)             vector (x, y, z)
    1 , I               .000000    .000000    .000000
    C2,C4,M2,M4        1.000000    .000000    .000000
    C2,C4,M2,M4         .000000   1.000000    .000000
    C2,C4,M2,M4         .000000    .000000   1.000000
    C2, M2              .707107    .707107    .000000
    C2, M2              .707107   -.707107    .000000
    C2, M2              .707107    .000000    .707107
    C2, M2              .707107    .000000   -.707107
    C2, M2              .000000    .707107    .707107
    C2, M2              .000000    .707107   -.707107
    C3, M3              .577350    .577350    .577350
    C3, M3              .577350    .577350   -.577350
    C3, M3              .577350   -.577350    .577350
    C3, M3              .577350   -.577350   -.577350
 
 
b)  face centered cubic lattice (lattice code = 2, -2).
 
  lattice vectors                    reciprocal lattice vectors
         x         y         z               x         y         z
  R1   0.00000   0.50000   0.50000    G1  -6.28319   6.28319   6.28319
  R2   0.50000   0.00000   0.50000    G2   6.28319  -6.28319   6.28319
  R3   0.50000   0.50000   0.00000    G3   6.28319   6.28319  -6.28319
 
  1 lattice basis vector
          x         y         z      atom/ion radius   nuclear charge
  r1   0.00000   0.00000   0.00000       0.3536            13  (Al)
 
  packing ratio =  0.74048
 
  48 symmetry operations :  see simple cubic lattice
 
 
c)  body centered cubic lattice (lattice code = 3, -3).
 
  lattice vectors                    reciprocal lattice vectors
         x         y         z               x         y         z
  R1  -0.50000   0.50000   0.50000    G1   0.00000   6.28319   6.28319
  R2   0.50000  -0.50000   0.50000    G2   6.28319   0.00000   6.28319
  R3   0.50000   0.50000  -0.50000    G3   6.28319   6.28319   0.00000
 
  1 lattice basis vector
          x         y         z      atom/ion radius   nuclear charge
  r1   0.00000   0.00000   0.00000       0.4330            26  (Fe)
 
  packing ratio =  0.68017
 
  48 symmetry operations :  see simple cubic lattice
 
 
d)  hexagonal closed packed lattice (lattice code = 4, -4) defined as
    hexagonal with 2 atoms per unit cell.
 
  lattice vectors                    reciprocal lattice vectors
         x         y         z               x         y         z
  R1   1.00000   0.00000   0.00000    G1   6.28319  -3.62760   0.00000
  R2   0.50000   0.86603   0.00000    G2   0.00000   7.25520   0.00000
  R3   0.00000   0.00000   1.63299    G3   0.00000   0.00000   3.84765
 
  2 lattice basis vectors
          x         y         z      atom/ion radius   nuclear charge
  r1   0.00000   0.00000   0.00000       0.5000            27  (Co)
  r2   0.50000   0.28870   0.81650       0.5000            27  (Co)
 
  packing ratio =  0.74048
 
  12 symmetry operations :  1x1, 3xC2, 2xC3, 4xM2, 2xM6
 
    type(s)             vector (x, y, z)
    1                   .000000    .000000    .000000
    C2                  .866025    .500000    .000000
    C2                  .866025   -.500000    .000000
    C2                  .000000   1.000000    .000000
    C3,M2,M6            .000000    .000000   1.000000
    M2                 1.000000    .000000    .000000
    M2                  .500000    .866025    .000000
    M2                  .500000   -.866025    .000000
 
 
e)  diamond lattice (lattice code = 5, -5) defined as face centered cubic
    with 2 atoms per unit cell.
 
  lattice vectors                    reciprocal lattice vectors
         x         y         z               x         y         z
  R1   0.00000   0.50000   0.50000    G1  -6.28319   6.28319   6.28319
  R2   0.50000   0.00000   0.50000    G2   6.28319  -6.28319   6.28319
  R3   0.50000   0.50000   0.00000    G3   6.28319   6.28319  -6.28319
 
  2 lattice basis vectors
          x         y         z      atom/ion radius   nuclear charge
  r1   0.00000   0.00000   0.00000       0.2165            6   (C)
  r2   0.25000   0.25000   0.25000       0.2165            6   (C)
 
  packing ratio =  0.34009
 
  24 symmetry operations :  1x1, 3xC2, 8xC3, 6xM2, 6xM4
 
    type(s)             vector (x, y, z)
    1                   .000000    .000000    .000000
    C2, M4             1.000000    .000000    .000000
    C2, M4              .000000   1.000000    .000000
    C2, M4              .000000    .000000   1.000000
    C3                  .577350    .577350    .577350
    C3                  .577350    .577350   -.577350
    C3                  .577350   -.577350    .577350
    C3                  .577350   -.577350   -.577350
    M2                  .707107    .707107    .000000
    M2                  .707107   -.707107    .000000
    M2                  .707107    .000000    .707107
    M2                  .707107    .000000   -.707107
    M2                  .000000    .707107    .707107
    M2                  .000000    .707107   -.707107
 
 
f)  sodium chloride lattice (lattice code = 6, -6) defined as face centered
    cubic with 2 atoms per unit cell. The radii ratio r(Na)/r(Cl) = 0.525
    is taken from Coulson's tables of ionic radii.
 
  lattice vectors                    reciprocal lattice vectors
         x         y         z               x         y         z
  R1   0.00000   0.50000   0.50000    G1  -6.28319   6.28319   6.28319
  R2   0.50000   0.00000   0.50000    G2   6.28319  -6.28319   6.28319
  R3   0.50000   0.50000   0.00000    G3   6.28319   6.28319  -6.28319
 
  2 lattice basis vectors
          x         y         z      atom/ion radius   nuclear charge
  r1   0.00000   0.00000   0.00000       0.1721            11  (Na)
  r2   0.00000   0.00000   0.50000       0.3279            17  (Cl)
 
  packing ratio =  0.67599
 
  48 symmetry operations :  see simple cubic lattice
 
 
g)  cesium chloride lattice (lattice code = 7) defined as simple cubic with 2
    atoms per unit cell. Ratio of radii r(Cs)/r(Cl) = 0.934 is taken from
    Coulson's tables of ionic radii.
 
  lattice vectors                    reciprocal lattice vectors
         x         y         z               x         y         z
  R1   1.00000   0.00000   0.00000    G1   6.28319   0.00000   0.00000
  R2   0.00000   1.00000   0.00000    G2   0.00000   6.28319   0.00000
  R3   0.00000   0.00000   1.00000    G3   0.00000   0.00000   6.28319
 
  2 lattice basis vectors
          x         y         z      atom/ion radius   nuclear charge
  r1   0.00000   0.00000   0.00000       0.4182            55  (Cs)
  r2   0.50000   0.50000   0.50000       0.4478            17  (Cl)
 
  packing ratio =  0.68255
 
  48 symmetry operations :  see simple cubic lattice
 
 
h)  cubic Zincblende lattice (lattice code = 8, -8) defined as face centered
    cubic with 2 atoms per unit cell. Ratio of radii r(Zn)/r(S) = 0.476 is
    taken from Coulson's tables of ionic radii.
 
  lattice vectors                    reciprocal lattice vectors
         x         y         z               x         y         z
  R1   0.00000   0.50000   0.50000    G1  -6.28319   6.28319   6.28319
  R2   0.50000   0.00000   0.50000    G2   6.28319  -6.28319   6.28319
  R3   0.50000   0.50000   0.00000    G3   6.28319   6.28319  -6.28319
 
  2 lattice basis vectors
          x         y         z      atom/ion radius   nuclear charge
  r1   0.00000   0.00000   0.00000       0.1396            30  (Zn)
  r2   0.25000   0.25000   0.25000       0.2934            16  (S)
 
  packing ratio =  0.46868
 
  24 symmetry operations :  see diamond lattice
 
 
i)  Graphite lattice (lattice code = 9, -9) defined as hexagonal with 4
    atoms per unit cell. A lattice constant ratio c/a = 2.72 is used as
    default.
 
  lattice vectors                    reciprocal lattice vectors
         x         y         z               x         y         z
  R1   1.00000   0.00000   0.00000    G1   6.28319  -3.62760   0.00000
  R2   0.50000   0.86603   0.00000    G2   0.00000   7.25520   0.00000
  R3   0.00000   0.00000   2.72000    G3   0.00000   0.00000   2.30999
 
  4 lattice basis vectors
          x         y         z      atom/ion radius   nuclear charge
  r1   0.00000   0.00000   0.00000       0.2887            6   (C)
  r2   0.50000   0.28870   0.00000       0.2887            6   (C)
  r3   0.00000   0.00000   1.36000       0.2887            6   (C)
  r4   1.00000   0.57740   1.36000       0.2887            6   (C)
 
  packing ratio =  0.17111
 
  12 symmetry operations :  see hexagonal closed packed lattice
 
 
 
 
B1) Bravais lattices
 
The following table gives, for each of the 14 Bravais lattices, all
parameters necessary to define the lattice and cartesian coordinates of all
lattice vectors R1, R2, R3 with lengths |R1| = a, |R2| = b, |R3| = c, and
angles &lt;(R1,R2) = v12, &lt;(R1,R3) = v13, &lt;(R2,R3) = v23. Further, we define
 
     cij = cos(vij) ,    sij = sin(vij) ,  i,j = 1, 2, 3.
 
 
a)  Triclinic-P lattice
 
    Lattice without symmetry (apart from inversion).
 
    Parameters a, b, c, v12, v13, v23 required.
 
       R1 =  a ( 1,      0,  0 )
       R2 =  b ( c12,  s12,  0 )
       R3 =  c ( c13, (c23-c13*c12)/s12,
                               sqrt(1-c12**2-c13**2-c23**2+c12*c13*c23)/s12 )
 
   2 symmetry operations :  1x1, 1xI
 
    type(s)             vector (x, y, z)
    1 , I               .000000    .000000    .000000
 
 
b)  Monoclinic-P lattice
 
    Three different angular cases:
 
    Case 1:   v23,  v13 = v12 = 90,    vector R1 perpendicular to R2, R3.
 
       Parameters a, b, c,  v23 required (all tree angles must be given).
 
          R1 =  a (   1,   0,   0 )
          R2 =  b (   0,   1,   0 )
          R3 =  c (   0, c23, s23 )
 
      4 symmetry operations :  1x1, 1xC2, 1xI, 1xM2
 
       type(s)             vector (x, y, z)
       1 , I               .000000    .000000    .000000
       C2,M2              1.000000    .000000    .000000
 
    Case 2:   v13,  v23 = v12 = 90,    vector R2 perpendicular to R1, R3.
 
       Parameters a, b, c,  v13 required (all tree angles must be given).
 
          R1 =  a (   1,   0,   0 )
          R2 =  b (   0,   1,   0 )
          R3 =  c ( c13,   0, s13 )
 
      4 symmetry operations :  1x1, 1xC2, 1xI, 1xM2
 
       type(s)             vector (x, y, z)
       1 , I               .000000    .000000    .000000
       C2,M2               .000000   1.000000    .000000
 
    Case 3:   v12,  v13 = v23 = 90,    vector R3 perpendicular to R1, R2.
 
       Parameters a, b, c,  v12 required (all tree angles must be given).
 
          R1 =  a (   1,   0,   0 )
          R2 =  b ( c12, s12,   0 )
          R3 =  c (   0,   0,   1 )
 
      4 symmetry operations :  1x1, 1xC2, 1xI, 1xM2
 
       type(s)             vector (x, y, z)
       1 , I               .000000    .000000    .000000
       C2,M2               .000000    .000000    1.000000
 
 
c)  Monoclinic-C lattice
 
    Three different angular cases:
 
    Case 1:   v23,  v13 = v12 = 90,    R1/R3 plane centered.
 
       Parameters a, b, c,  v23 required (all tree angles must be given).
 
          R1 =  a ( 1/2, -c23/2, -s23/2 )
          R2 =  b (   0,      1,      0 )
          R3 =  c ( 1/2,  c23/2,  s23/2 )
 
      4 symmetry operations :  see monoclinic-P lattice
 
    Case 2:   v13,  v23 = v12 = 90,    R1/R2 plane centered.
 
       Parameters a, b, c,  v13 required (all tree angles must be given).
 
          R1 =  a (  .5,  .5,   0 )
          R2 =  b ( -.5,  .5,   0 )
          R3 =  c ( c13,   0, s13 )
 
      4 symmetry operations :  see monoclinic-P lattice
 
    Case 3:   v12,  v13 = v23 = 90,    vector R3 perpendicular to R1, R2.
 
       Parameters a, b, c,  v12 required (all tree angles must be given).
 
          R1 =  a (      1,      0,    0 )
          R2 =  b (  c12/2,  s12/2,  1/2 )
          R3 =  c ( -c12/2, -s12/2,  1/2 )
 
      4 symmetry operations :  see monoclinic-P lattice
 
 
d)  Orthorhombic-P lattice
 
    Vectors R1, R2, R3 perpendicular to each other.
 
    Parameters a, b, c required.
 
       R1 =  ( a,  0,  0 )
       R2 =  ( 0,  b,  0 )
       R3 =  ( 0,  0,  c )
 
   8 symmetry operations :  1x1, 3xC2, 1xI, 3xM2
 
    type(s)             vector (x, y, z)
    1 , I               .000000    .000000    .000000
    C2,M2              1.000000    .000000    .000000
    C2,M2               .000000   1.000000    .000000
    C2,M2               .000000    .000000   1.000000
 
 
e)  Orthorhombic-C lattice
 
    Vectors R1, R2, R3 perpendicular to each other. Plane R1/R2 is centered
    rectangular.
 
    Parameters a, b, c required.
 
       R1 =  1/2 ( a, -b,  0 )
       R2 =  1/2 ( a,  b,  0 )
       R3 =      ( 0,  0,  c )
 
   8 symmetry operations :  see orthorhombic-P lattice
 
 
f)  Orthorhombic-I lattice
 
    Vectors R1, R2, R3 perpendicular to each other. One additional atom in
    cell center.
 
    Parameters a, b, c required.
 
       R1 =  1/2 (  a, -b,  c )
       R2 =  1/2 (  a,  b, -c )
       R3 =  1/2 ( -a,  b,  c )
 
   8 symmetry operations :  see orthorhombic-P lattice
 
 
g)  Orthorhombic-F lattice
 
    Vectors R1, R2, R3 perpendicular to each other. Planes R1/R2, R1/R3, R2/R3
    are centered rectangular.
 
    Parameters a, b, c required.
 
       R1 =  1/2 ( a,  0,  c )
       R2 =  1/2 ( a,  b,  0 )
       R3 =  1/2 ( 0,  b,  c )
 
   8 symmetry operations :  see orthorhombic-P lattice
 
 
h)  Tetragonal-P lattice
 
    Vectors R1, R2, R3 perpendicular to each other with |R1| = |R2| = a.
 
    Parameters a, c required.
 
       R1 =  ( a,  0,  0 )
       R2 =  ( 0,  a,  0 )
       R3 =  ( 0,  0,  c )
 
  16 symmetry operations :  1x1, 5xC2, 2xC4, 1xI, 5xM2, 2xM4
 
    type(s)             vector (x, y, z)
    1 , I               .000000    .000000    .000000
    C2,M2              1.000000    .000000    .000000
    C2,M2               .000000   1.000000    .000000
    C2,C4,M2,M4         .000000    .000000   1.000000
    C2,M2               .707107    .707107    .000000
    C2,M2               .707107   -.707107    .000000
 
 
i)  Tetragonal-I lattice
 
    Vectors R1, R2, R3 perpendicular to each other with |R1| = |R2| = a.
    One additional atom in cell center.
 
    Parameters a, c required.
 
       R1 =  1/2 (  a, -a,  c )
       R2 =  1/2 (  a,  a, -c )
       R3 =  1/2 ( -a,  a,  c )
 
  16 symmetry operations :  see tetragonal-P lattice
 
 
j)  Hexagonal-P lattice
 
    Vectors R3 perpendicular R1, R2 with |R1| = |R2| = a  and  v12 = 120.
 
    Parameters a, c required.
 
       R1 =  a ( q, -1/2,  0 )
       R2 =  a ( 0,    1,  0 )      q = sqrt(3/4)
       R3 =  c ( 0,    0,  1 )
 
  24 symmetry operations :  1x1, 7xC2, 2xC3, 2xC6, 1xI, 7xM2, 2xM3, 2xM6
 
    type(s)             vector (x, y, z)
    1 , I               .000000    .000000    .000000
    C2,C3,C6,M2,M3,M6   .000000    .000000   1.000000
    C2, M2             1.000000    .000000    .000000
    C2, M2              .866025    .500000    .000000
    C2, M2              .866025   -.500000    .000000
    C2, M2              .500000    .866025    .000000
    C2, M2              .500000   -.866025    .000000
    C2, M2              .000000   1.000000    .000000
 
 
k)  Trigonal-R lattice
 
    Vectors R1, R2, R3 form an equilateral rhombohedron where
    |R1| = |R2| = |R3| = a  and  v12 = v13 = v23 .
 
    Parameters a, v12 &lt; 120 required.
 
       R1 =  a (    p12,       0, q12 )         p12 = sqrt( (1-c12)*2/3 )
       R2 =  a ( -p12/2,   p12*q, q12 )           q = sqrt(3/4)
       R3 =  a ( -p12/2,  -p12*q, q12 )         q12 = sqrt( 1- p12**2 )
 
  12 symmetry operations :  1x1, 3xC2, 2xC3, 1xI, 3xM2, 2xM3
 
    type(s)             vector (x, y, z)
    1 , I               .000000    .000000    .000000
    C3, M3              .000000    .000000   1.000000
    C2, M2              .000000   1.000000    .000000
    C2, M2              .866025    .500000    .000000
    C2, M2              .866025   -.500000    .000000
 
 
l)  Cubic-P lattice
 
    Vectors R1, R2, R3 perpendicular to each other with
    |R1| = |R2| = |R3| = a.
 
    Parameter a required.
 
       R1 =  a ( 1,  0,  0 )
       R2 =  a ( 0,  1,  0 )
       R3 =  a ( 0,  0,  1 )
 
  48 symmetry operations :  1x1, 9xC2, 8xC3, 6xC4, 1xI, 9xM2, 8xM3, 6xM4
 
    type(s)             vector (x, y, z)
    1 , I               .000000    .000000    .000000
    C2,C4,M2,M4        1.000000    .000000    .000000
    C2,C4,M2,M4         .000000   1.000000    .000000
    C2,C4,M2,M4         .000000    .000000   1.000000
    C2, M2              .707107    .707107    .000000
    C2, M2              .707107   -.707107    .000000
    C2, M2              .707107    .000000    .707107
    C2, M2              .707107    .000000   -.707107
    C2, M2              .000000    .707107    .707107
    C2, M2              .000000    .707107   -.707107
    C3, M3              .577350    .577350    .577350
    C3, M3              .577350    .577350   -.577350
    C3, M3              .577350   -.577350    .577350
    C3, M3              .577350   -.577350   -.577350
 
 
m)  Cubic-I (bcc) lattice
 
    Vectors R1, R2, R3 perpendicular to each other with
    |R1| = |R2| = |R3| = a. One additional atom in cell center.
 
    Parameter a required.
 
       R1 =  a/2 (  1,  -1,   1 )
       R2 =  a/2 (  1,   1,  -1 )
       R3 =  a/2 ( -1,   1,   1 )
 
  48 symmetry operations :  see cubic-P lattice
 
 
n)  Cubic-f (fcc) lattice
 
    Vectors R1, R2, R3 perpendicular to each other with
    |R1| = |R2| = |R3| = a. Planes R1/R2, R1/R3, R2/R3 are centered square.
 
    Parameter a required.
 
       R1 =  a/2 (  1,  0,  1 )
       R2 =  a/2 (  1,  1,  0 )
       R3 =  a/2 (  0,  1,  1 )
 
  48 symmetry operations :  see cubic-P lattice
 
 
 
The following table lists lattices and lattice constants a (in Angstroms) for
the most common elements as taken from R. W. G. Wyckoff, "Crystal
Structures", Interscience, New York 1963. Nearly closed packed hexagonal
lattices are denoted by hcp and the two lattice constants a/c of the
hexagonal lattice are given.
 
     charge   lattice    a (a/c)           charge   lattice    a (a/c)
    ------------------------------        ------------------------------
      2 (He)   hcp      3.57/5.83          47 (Ag)   fcc      4.09
      3 (Li)   bcc      3.49               48 (Cd)   hcp      2.98/5.62
      4 (Be)   hcp      2.29/3.58          50 (Sn)   diamond  6.49
      6 (C )   diamond  3.57               54 (Xe)   fcc      6.20
     10 (Ne)   fcc      4.43               55 (Cs)   bcc      6.05
     11 (Na)   bcc      4.23               56 (Ba)   bcc      5.02
     12 (Mg)   hcp      3.21/5.21          57 (La)   fcc      5.30
     13 (Al)   fcc      4.05               57 (La)   hcp      3.75/6.07
     14 (Si)   diamond  5.43               58 (Ce)   fcc      5.16
     18 (Ar)   fcc      5.26               58 (Ce)   hcp      3.65/5.96
     19 (K )   bcc      5.23               59 (Pr)   fcc      5.16
     20 (Ca)   fcc      5.58               59 (Pr)   hcp      3.67/5.92
     21 (Sc)   fcc      4.54               60 (Nd)   hcp      3.66/5.90
     21 (Sc)   hcp      3.31/5.27          64 (Gd)   hcp      3.64/5.78
     22 (Ti)   hcp      2.95/4.69          65 (Tb)   hcp      3.60/5.69
     23 (V )   bcc      3.02               66 (Dy)   hcp      3.59/5.65
     24 (Cr)   bcc      2.88               67 (Ho)   hcp      3.58/5.62
     26 (Fe)   bcc      2.87               68 (Er)   hcp      3.56/5.59
     27 (Co)   hcp      2.51/4.07          69 (Tm)   hcp      3.54/5.55
     27 (Co)   fcc      3.55               70 (Yb)   fcc      5.49
     28 (Ni)   fcc      3.52               71 (Lu)   hcp      3.50/5.55
     29 (Cu)   fcc      3.61               72 (Hf)   hcp      3.20/5.06
     30 (Zn)   hcp      2.66/4.95          73 (Ta)   bcc      3.31
     32 (Ge)   diamond  5.66               74 (W )   bcc      3.16
     36 (Kr)   fcc      5.72               75 (Re)   hcp      2.76/4.46
     37 (Rb)   bcc      5.59               76 (Os)   hcp      2.74/4.32
     38 (Sr)   fcc      6.08               77 (Ir)   fcc      3.84
     39 (Y )   hcp      3.65/5.73          78 (Pt)   fcc      3.92
     40 (Zr)   bcc      3.61               79 (Au)   fcc      4.08
     40 (Zr)   hcp      3.23/5.15          81 (Tl)   bcc      3.88
     41 (Nb)   bcc      3.30               81 (Tl)   hcp      3.46/5.53
     42 (Mo)   bcc      3.15               82 (Pb)   fcc      4.95
     44 (Ru)   hcp      2.70/4.28          90 (Th)   fcc      5.08
     45 (Rh)   fcc      3.80               94 (Pu)   fcc      4.64
     46 (Pd)   fcc      3.89
 
 
 
 

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